#8202256

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#38025	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,5)(3,7,8,6)(9,10)(11,18)(12,22)(13,17)(14,15)(16,19)(20,21)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(2,5)(6,7)(10,15)(11,19)(12,17)(21,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(10,15)(13,22)(16,18)(21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,18)(17,23)(22,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,4,8)(2,6,5,7)(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109452);